Problem Solving

100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors?

A)6
B)16
C)17
D)33
E)84

Why OAB why not C

j_shreyans wrote:100 people are attending a newspaper conference. 45 of them are writers and more than 38 are editors. Of the people at the conference, x are both writers and editors and 2x are neither. What is the largest possible number of people who are both writers and editors?

A)6
B)16
C)17
D)33
E)84

Total = writers + editors - both + neither.

The big idea is to SUBTRACT the overlap.
When we count the total number of writers and the total number of editors, the OVERLAP -- everyone who is BOTH a writer and an editor -- is counted TWICE.
Thus, the people who are in BOTH groups must be subtracted from the total so that they are not double-counted.

In the equation above:
Total = 100.
Writers = 45.
Both = x.
Neither = 2x.

Plugging these values into the equation:
100 = 45 + editors - x + 2x
55 = editors + x.

In order to MAXIMIZE x, we need to MINIMIZE the number of editors.
Since more than 38 of the people are editors, the minimum number of editors = 39:
55 = 39 + x
x = 16.

The correct answer is B.